The claim is false.
According to the National Statistics Authority
The UK’s official gross contribution for 2014 before the application of the rebate was £19.1 billion. As I have made clear previously, this is not an amount of money that the UK pays to the EU each year. The full £19.1 billion is not a net contribution.
Letter from Sir Andrew Dilnot to Dominic Cummings
This is also confirmed by HM Treasury.
The problem with the claim is that first of all, a discount was negotiated by the UK in the 80's, so while the official membership cost is 19 billions, that money is not what is actually paid at all by the country. Secondly, part of those funds are returned later on to the country in form of subsidies.
A nice recap of this is provided by Full Fact which was further verified by In Facts.
Further debunking can be easily found on the Financial Times
Britain does not contribute a net £350m a week to the EU budget. Savings will not go automatically to the National Health Service.
and the Economist
Promises of more money rely on the
claim that leaving the EU will save Britain’s
EU budget payments of £350m ($510m) a
week. But when the rebate and EU spending
in Britain is accounted for, the net payment
is only £120m a week.